Simplify sin(arctan(x))

Problem

sin(arctan(x))

Solution

1. Identify the inner function as an angle θ=arctan(x) which implies tan(θ)=x

2. Represent the relationship using a right triangle where the side opposite to θ is x and the side adjacent to θ is 1

3. Calculate the hypotenuse of the triangle using the Pythagorean theorem: √(,x2+1)=√(,x2+1)

4. Determine the sine of the angle θ using the ratio of the opposite side to the hypotenuse.

5. Substitute the ratio back into the expression sin(θ)=x/√(,x2+1)

Final Answer

sin(arctan(x))=x/√(,x2+1)

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